16 APPLICATIONS: I. INTEGRATION 199 16. APPLICATIONS: I. INTEGRATION WITH RESPECT TO A COMPLEX MEASURE (13.16.1) Let n, v be two positive measures on X. (i) We have (/JL + v)* = /r* -f v*. (ii) For a function f to be (JLL + v)~integrable (resp. GU -f ^-measurable) it is necessary and sufficient that f should be \i-integrable and v-integrable (resp. ^-measurable and v-measurable\ and in that case we have (13.16.1.1) f/dfo + v^ \fdn+ |/dv. We may write \JL = g • (^ + v) and v = h - (ft + v), where g and h are ^0 and locally (\JL + v)-integrable ((13.15.1) and (13.15.3)). Since ju -f v = (#•4- h) - (n + v), it follows that #(x) + /*(*) = 1 almost everywhere with respect to ^ + v. Put A = ^ + v. To prove (i), we have to show that f*/dA= |*/0<tt+ f for all functions/^ 0. Clearly J*/dA ^ J*/^ dJ, + J*/A ^- Also there is no r* loss of generality in assuming that /dX < + oo, in which case there exists a decreasing sequence (un) of ,1-integrable functions belonging to ./, such that f^un for all n and |*f dh = inf \un dl. Since \un dh = \ungdk-\- \unhdh we obtain the desired inequality by passing to the limit. Assertion (ii) then follows from (13.14.2) and (13.14.3). Now let A be a complex measure on X. A complex-valued function/on X is said to be h-integrable if it is |A|-integrable. If we put Xl = 0lk and A2 = «/A, it follows from (13.3.7) and (13.16.1) that/is A-integrable if and only if/is ^-integrable and I2"integrable. Since by (13.15.3.1) we have \X^\ = Aj" + ^ and |12I = ^2" + &2> fr follows that/is A-integrable if and only if/is integrable with respect to each of the four positive measures Aj", Af, X%, ^2 • The integral off with respect to A is then defined to be the complex number (13.16.2) \fdJi = /! dtf - /d^" + i \fd^ - i.9.13). Hence by (13.14.3) /is